It is widely known that the algebaric closure of the p-adic $p$-adic completion $\mathbb{Q_p}$ \mathbb{Q}_p$ of $\mathbb{Q}$ isn't complete anymore. It's completion is complete and known as $\mathbb{C_p}$. \mathbb{C}_p$.
I have read in a book about non-archimedean analysis that in this case the process ends, which means that $\mathbb{C_p}$ \mathbb{C}_p$ is also algebraically closed.
My question is: is there an example of a field K, in which the algebraic closure $K^alg$ K^{alg}$ isn't complete, and the completion of $K^alg$ K^{alg}$ isn't algebraically closed ? And how do I construct such an example.
